A novel improved domain material point method for the cell-crossing instability problem

This paper introduces an improved domain material point method (IDMPM) by scrutinizing the integration error of internal forces and considering the algorithmic properties of the standard material point method (MPM). The traditional uniform shape function is enhanced by designing internal and boundary elements separately, thus providing a distinct expression with an influence domain within the interval [1.0, 2.0]. In this method, the peak value of the form function is located at the grid node, and the expression of the form function is provided for the boundary element. This approach can help avoid the mass and momentum nonconservation issues of Generalized Interpolation Material Point Method (GIMP) and B-spline Material Point Method (BSMPM). Expressing the form function in polynomial form can significantly enhance the computational efficiency of the program. The IDMPM is appraised through a series of numerical examples that address small elastic deformations, dynamic large deformations, and extremely large deformations under diverse dimensions and boundary conditions. The outcomes demonstrate that the IDMPM outperforms the standard MPM in terms of accuracy, convergence speed, and energy conservation properties. Notably, when compared with the B-spline material point method (BSMPM), IDMPM demonstrates superior calculation accuracy and efficiency due to the boundary element shape functions employed.